Question

what is the difference in finding the basis of a subset and a basis of a null space? I just need some explanation to what the difference between a subspace and a null space is, I think that would help me understand. Thanks!

Answer 1

1/ subset is not a subspace--> subset doesn't have a basis
2/a sub-space is a subset of the whole space. It has the same property as the whole space but less elements.
3/ Null space is set of elements which makes the define operator on the WHOLE space =0
You can imagine the R^3 space. Its basis is {(1,0,0),(0,1,0), (0,0,1)} dimension 3 like x, y,z coordinate. It is "built" by xy, yz, xz plans. So, each of xy, yz, xz are subspace of R^3
Let define an operator on R^3, write f = x + y -z
what make f =0? if x =1, z= 1 and y =0, then f=0 , right? so?? the vector on R^3 = (1, 0,1) and its multiple will make the operator =0. Gather all those kind of vectors, you have null space
Hopefully It helps you understand more about the stuff